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Chapter 5: Problem 29

Smoke alarms statistics National estimates of reported fires derived from the National Fire Incident Reporting System (NFIRS) and the National Fire Protection Association's (NFPA's) fire department survey show that in \(2009-2013,38 \%\) of home fire deaths occurred in homes with no smoke alarms, and \(21 \%\) of home fire deaths were caused by fires in which smoke alarms were present but failed to operate. Let \(D\) denote \\{home fire death \(\\}, P\) denote \\{Smoke alarm is present\\}, and let F denote \\{Failed to oper- ate\\}. Using events and their complements, identify each of the two given probabilities as a conditional probability.

Short Answer

Expert verified
The conditional probabilities are: 1. \(P(\text{not } P \,|\, D) = 0.38\) 2. \(P(F \,|\, D \text{ and } P) = 0.21\).

Step by step solution

01

Understand the Problem

We need to identify each of the given percentages as conditional probabilities related to home fire deaths and smoke alarm conditions. Define the necessary events: \(D\) means a home fire death occurs, \(P\) means a smoke alarm is present, and \(F\) means the alarm fails to operate.
02

Identify Probabilities

According to the problem, the probabilities are as follows: - 38% of home fire deaths occurred in homes with no smoke alarms. - 21% of home fire deaths occurred in homes where smoke alarms failed to operate. These can be expressed in terms of conditional probabilities.
03

Define Conditional Probability 1

The first statement means: once we know a death occurred \( \{D\} \), there is a 38% probability that no smoke alarm was present \( \{D \text{ and not } P\} \). This is the conditional probability: \[ P(\text{not } P \,|\, D) = 0.38 \]
04

Define Conditional Probability 2

The second statement means: once we know a death occurred \( \{D\} \), there is a 21% probability that a smoke alarm was present but failed to operate \( \{D \text{ and } P \text{ and } F\} \). This is another conditional probability: \[ P(F \,|\, D \text{ and } P) = 0.21 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Home Fire Safety Statistics
Home fire safety statistics provide crucial insights into the occurrences and circumstances of residential fires. Between 2009 and 2013, data from the National Fire Incident Reporting System (NFIRS) and the National Fire Protection Association (NFPA) indicated some eye-opening trends. Specifically, it was observed that 38% of home fire deaths happened in residences without smoke alarms. This statistic highlights the fundamental importance of having smoke alarms installed and functioning.

Additionally, 21% of deaths occurred under circumstances where smoke alarms were present but failed to operate correctly. This points out the crucial aspect of not only installing smoke alarms but ensuring their consistent maintenance. Regular checks and ensuring batteries are functional can significantly affect these statistics.

These numbers collectively emphasize proactive measures for home fire safety, illustrating the need for awareness and preparedness in preventing tragedies.
Evaluating Smoke Alarm Effectiveness
Smoke alarms are essential tools in reducing the risk of deaths in household fires. Their effectiveness is primarily measured by their presence and operability during fire incidents. Present yet faulty smoke alarms accounted for 21% of home fire deaths. This indicates a significant lapse in safety measures that can potentially be addressed with better maintenance practices.
  • Regularly test smoke alarms to ensure they function as expected.
  • Replace batteries at least once a year or as soon as they emit a low-battery warning.
  • Consider interconnected smoke alarms, which sound all alarms in the system when one detects smoke, increasing the odds of early detection.
Ensuring regularly scheduled maintenance can greatly enhance their reliability. Properly installed and maintained smoke alarms are a proven defense, significantly lowering the risk of fire fatalities.
Basics of Event Probability Analysis
Event probability analysis is a fundamental statistical approach used to determine the likelihood of certain outcomes, particularly in assessing risk factors and safety measures. For this context, it鈥檚 about recognizing and interpreting the likelihood of events like fires occurring in homes with varying conditions for smoke alarms.
  • Conditional Probability: This helps to determine the probability of an event occurring, provided another related event has already occurred. For example, we are interested in understanding: given a fire death occurred, what is the likelihood that no smoke alarm was present?
  • Interpreting Percentages as Probabilities: The 38% figure translates to a significant possibility that a home fire death happens in the absence of alarms, calculated as a conditional probability where alarms are not present despite fire deaths occurring.
  • Understanding Complements: Knowing the probability of a smoke alarm's failure conditionally helps set strategies to mitigate that by understanding and reducing occurrences where they don鈥檛 operate.
This analytical method provides insights that facilitate strategic decisions and policy-making geared towards fire prevention, reinforcing smoke alarm presence and maintenance.

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